1. Find the Median of a Stream:

Given a continuous stream of numbers, design a data structure that efficiently finds the median of the numbers seen so far.

Algorithm:

1. Initialize an empty max-heap (lowerHeap) to store the smaller half of the numbers and an empty min-heap (higherHeap) to store the larger half of the numbers.
2. Whenever a new number is received from the stream:
   • If both heaps are empty, add the number to the max-heap.
   • If the number is smaller than the median (the root of the max-heap), add it to the max-heap; otherwise, add it to the min-heap.
3. After adding a number, ensure that the heaps are balanced, with the size difference between them at most 1. If the difference exceeds 1, perform the following steps:
   • If the max-heap (lowerHeap) has more elements, remove the root element (the maximum) from the max-heap and add it to the min-heap.
   • If the min-heap (higherHeap) has more elements, remove the root element (the minimum) from the min-heap and add it to the max-heap.
4. To find the median, check the sizes of the heaps:
   • If the heaps have the same size, the median is the average of the roots of both heaps.
   • If the max-heap (lowerHeap) has more elements, the median is the root of the max-heap.
   • If the min-heap (higherHeap) has more elements, the median is the root of the min-heap.

import heapq

class MedianFinder:
    def __init__(self):
        self.lowerHeap = []  # max-heap
        self.higherHeap = []  # min-heap

    def addNum(self, num):
        if len(self.lowerHeap) == len(self.higherHeap):
            # Add the number to the max-heap (lowerHeap) and negate it to simulate a max-heap behavior
            heapq.heappush(self.lowerHeap, -num)
        else:
            # Add the number to the min-heap (higherHeap)
            heapq.heappush(self.higherHeap, num)

        # Balance the heaps if necessary
        if self.higherHeap and -self.lowerHeap[0] > self.higherHeap[0]:
            # Swap the root elements of both heaps
            heapq.heappush(self.higherHeap, -heapq.heappop(self.lowerHeap))
            heapq.heappush(self.lowerHeap, -heapq.heappop(self.higherHeap))

    def findMedian(self):
        if len(self.lowerHeap) == len(self.higherHeap):
            # Both heaps have the same size, so the median is the average of the roots
            return (-self.lowerHeap[0] + self.higherHeap[0]) / 2.0
        else:
            # Max-heap (lowerHeap) has more elements, so the median is the root of the max-heap
            return -self.lowerHeap[0]

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2. Sliding Window Median:

Given an array of integers and a window size, find the median of each window as it slides through the array.

Algorithm:

1. Define two data structures, a max-heap (also known as a max-priority queue) and a min-heap (min-priority queue). The max-heap will store the smaller half of the numbers in the window, while the min-heap will store the larger half.

2. Initialize the heaps and create a window of the given size. Insert the first k elements of the array into the heaps. If the window size is odd, insert the first k/2 + 1 elements into the max-heap and the rest into the min-heap. If the window size is even, insert the first k/2 elements into both heaps.

3. For each subsequent window, do the following:

   • Remove the element that is going out of the window from the heaps.
   • Insert the new element coming into the window into the appropriate heap.
   • Balance the heaps by ensuring that the difference in the number of elements between the max-heap and min-heap is at most 1. If the difference becomes larger, move the root element of the larger heap to the other heap.

4. Find the median for each window:

   • If the window size is odd, the median is the root element of the max-heap.
   • If the window size is even, the median is the average of the root elements of both heaps.

5. Repeat step 3 and step 4 until you have processed all windows in the array.

void insert(int val, multiset<int>&low, multiset<int>&high,int &k){
    int a = *low.rbegin(); // current median
    if (a<val){
        high.insert(val);
        if (high.size()>k/2){
            low.insert(*high.begin());
            high.erase(high.find(*high.begin()));
        }
    } else {
        low.insert(val);
        if (low.size()>(k+1)/2){
            high.insert(*low.rbegin());
            low.erase(low.find(*low.rbegin()));
        }
    }
 
}
void erase(int val, multiset<int>&low, multiset<int>&high){
	if (high.find(val) != high.end()) high.erase(high.find(val));
	else low.erase(low.find(val));
	if (low.empty()) {
		low.insert(*high.begin());
		high.erase(high.find(*high.begin()));
	}
}
 
signed main()
{ 
    multiset<int>low,high;
    int n,k;
    cin>>n>>k;
    vector < int > a(n,0);
    for (int i = 0; i < n; i++) cin>>a[i];
    low.insert(a[0]);
    for (int i = 1; i < k; i++) insert(a[i],low,high,k);
    cout<<*low.rbegin()<<" ";
    for (int i = k; i < n; i++) {
        if (k==1){
            insert(a[i],low,high,k);
            erase(a[i-1],low,high);
        }
        else{
            erase(a[i-k],low,high);
            insert(a[i],low,high,k);
        }
        cout<<*low.rbegin()<<" ";
    }
    cout<<endl;
 
    return 0;
}

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3.Kth Smallest Element in a Sorted Matrix:

Given a sorted matrix of integers, find the Kth smallest element.

• Binary Search Time Complexity : $O(K log N)$
• Bucket Sort Time Complexity: $O(n)$
• Heap Time Complexity: $O(N log K)$

Algorithm:

1. Initialize a min-heap data structure.
2. Iterate over the first row of the matrix and insert each element into the min-heap along with its corresponding row and column indices.
3. Start a loop that runs K times:
   • Extract the minimum element (smallest) from the min-heap. This element will be the Kth smallest element so far.
   • Check if there is an element in the next column of the current row. If so, insert it into the min-heap.
   • Increment the row index and insert the element from the next row in the same column into the min-heap.
4. After the loop finishes, the extracted element from the min-heap will be the Kth smallest element.

import heapq

def kthSmallest(matrix, k):
    if not matrix or not matrix[0]:
        return None
    
    rows, cols = len(matrix), len(matrix[0])
    min_heap = []
    
    # Insert elements from the first row into the min-heap
    for j in range(cols):
        heapq.heappush(min_heap, (matrix[0][j], 0, j))
    
    # Perform K iterations
    for _ in range(k):
        element, row, col = heapq.heappop(min_heap)
        
        # If there is a valid element in the next column of the current row, insert it into the min-heap
        if col + 1 < cols:
            heapq.heappush(min_heap, (matrix[row][col + 1], row, col + 1))
        
        # If it's not the last row, insert the element from the next row in the same column
        if row + 1 < rows:
            heapq.heappush(min_heap, (matrix[row + 1][col], row + 1, col))
    
    return element

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4. Find K Pairs with Smallest Sums:

Given two sorted arrays, find the K pairs with the smallest sums.

Required Time Complexity : $O(k * log(min(k, n)))$

Algorithm:

1. Create a min-heap data structure to store the pairs based on their sums. In Python, you can use the heapq module for this purpose.
2. Initialize an empty min-heap and push the first k pairs into the heap, sorted by their sums. You can compute the sums of the pairs and push them onto the heap with the pair itself.
3. Iterate through the remaining pairs, starting from index k. For each pair, calculate its sum and compare it with the root of the min-heap (the pair with the smallest sum so far).
4. If the sum is smaller than the root's sum, remove the root from the heap and push the current pair onto the heap.
5. Continue this process until you have processed all pairs. At the end, the k pairs with the smallest sums will be stored in the min-heap.
6. Extract the k pairs from the min-heap in ascending order of their sums. You can use the heappop function to remove the root of the heap repeatedly until you have extracted k pairs.

import heapq

def kSmallestPairs(nums1, nums2, k):
    if not nums1 or not nums2:
        return []
    
    heap = []
    for i in range(min(k, len(nums1))):
        heapq.heappush(heap, (nums1[i] + nums2[0], i, 0))
    
    pairs = []
    while k > 0 and heap:
        _, i, j = heapq.heappop(heap)
        pairs.append([nums1[i], nums2[j]])
        if j + 1 < len(nums2):
            heapq.heappush(heap, (nums1[i] + nums2[j+1], i, j+1))
        k -= 1
    
    return pairs

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5. Minimum Cost to Hire K Workers:

Given a list of workers with their respective qualities and minimum wage expectations, find the minimum cost to hire K workers such that the ratio of their total quality to total wage is minimized.

This problem is a variant of the classical "Minimum Cost Maximum Flow" problem. This problem can be solved using a combination of sorting and a priority queue.

Algorithm:

1. Create a list of workers, each containing their quality and minimum wage expectations.
2. Sort the workers in ascending order based on their wage/quality ratio. This step ensures that the workers with the lowest ratio will be considered first.
3. Initialize variables:
   • Initialize a priority queue to store the top K workers based on their qualities.
   • Initialize a variable to track the sum of the qualities of the workers in the priority queue.
4. Traverse the sorted list of workers:

• For each worker, calculate their wage/quality ratio.
  • Add the worker to the priority queue.
  • Add their quality to the sum of qualities.
  • If the priority queue size exceeds K, remove the worker with the highest quality from the queue and subtract their quality from the sum.

5. Calculate the total cost by multiplying the sum of qualities by the ratio of the last worker removed from the priority queue.
6. Return the total cost, which represents the minimum cost to hire K workers while minimizing the ratio of their total quality to total wage.

import heapq

def mincostToHireWorkers(qualities, wages, K):
    workers = sorted([(q, w) for q, w in zip(qualities, wages)], key=lambda x: x[1]/x[0])
    heap = []
    sum_qualities = 0
    min_cost = float('inf')

    for q, w in workers:
        heapq.heappush(heap, -q)
        sum_qualities += q

        if len(heap) > K:
            sum_qualities += heapq.heappop(heap)

        if len(heap) == K:
            min_cost = min(min_cost, sum_qualities * w / q)

    return min_cost

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Open Problem : Find Median in a Large File

This problem statement might not be complete

Given a large file containing a sorted list of integers that cannot fit into memory and are split into chunks, find the median of the numbers efficiently.

Kindly Contact me to contribute a solution.

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Other Problems :

• Huffman coding.
• Check if a binary tree is a heap.